Friction: Static and Kinetic Forces | 摩擦力：静摩擦与动摩擦

📚 Friction: Static and Kinetic Forces | 摩擦力：静摩擦与动摩擦

Friction is a contact force that opposes relative motion between two surfaces. In A-Level Mechanics, understanding friction is essential for analysing forces, equilibrium and motion on rough surfaces.

摩擦力是一种阻碍两个表面相对运动的接触力。在 A-Level 力学中，理解摩擦力对于分析粗糙表面上的力、平衡和运动至关重要。

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1. What Is Friction? | 什么是摩擦力？

Friction arises when two solid surfaces slide or attempt to slide over one another. It acts parallel to the contact plane and opposite to the direction of relative motion or the tendency to move.

当两个固体表面相互滑动或有滑动趋势时，就会产生摩擦力。它平行于接触面作用，方向与相对运动或运动趋势相反。

At the microscopic level, friction is caused by surface irregularities and adhesive bonds between contact points. Even surfaces that appear smooth have tiny peaks and valleys that interlock during sliding.

从微观角度看，摩擦力是由表面粗糙不平以及接触点之间的黏附作用引起的。即使看起来光滑的表面，在滑动时也会有微小的凸起和凹谷相互卡住。

Friction is a resistive force, but it is not always negative in effect. It enables walking, driving and holding objects. In many mechanical systems, however, friction causes wear and energy loss.

摩擦力是一种阻力，但其作用并不总是负面的。它使行走、驾驶和抓握物体成为可能。但在许多机械系统中，摩擦会导致磨损和能量损失。

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2. Static vs Kinetic Friction | 静摩擦与动摩擦

Static friction acts when an object is at rest and prevents it from starting to move. It can vary from zero up to a maximum value, called the limiting friction. Kinetic friction acts once the object is sliding and has a roughly constant magnitude for given surfaces.

静摩擦力作用于静止物体，阻止它开始运动。它可以从零变化到最大值，即极限摩擦。动摩擦力在物体滑动后起作用，对给定的表面，其大小近似恒定。

In most cases, the maximum static friction is slightly greater than kinetic friction. This is why it often takes more force to start an object moving than to keep it moving at constant speed.

在大多数情况下，最大静摩擦力略大于动摩擦力。这就是为什么让物体开始运动所需的力通常比保持它匀速运动所需的力更大。

For Edexcel A-Level, we normally assume that the coefficient of friction is the same for static and kinetic cases unless a question specifies otherwise.

在 Edexcel A-Level 中，除非题目另有说明，我们通常假设静摩擦和动摩擦的摩擦系数相同。

Static friction can prevent motion even when an applied force is present, as long as the applied force does not exceed the limiting value. Kinetic friction, by contrast, acts only while relative sliding occurs.

只要外力不超过极限值，静摩擦即使在外力存在时也能阻止运动。相比之下，动摩擦只在发生相对滑动时才起作用。

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3. The Coefficient of Friction | 摩擦系数

The coefficient of friction, μ, is a dimensionless number that describes the ratio of the friction force F to the normal reaction force R between two surfaces.

摩擦系数 μ 是一个无量纲的数，描述两个表面之间摩擦力 F 与法向反作用力 R 的比值。

F = μR

This equation applies to kinetic friction exactly and to the maximum static friction. The normal reaction R is the contact force perpendicular to the surfaces, often equal to the weight component acting perpendicular to the plane.

该方程精确适用于动摩擦，也适用于最大静摩擦。法向反作用力 R 是垂直于表面的接触力，通常等于垂直作用于平面的重力分量。

Typical values of μ range from about 0.01 for ice on ice to more than 1.0 for rubber on dry concrete. A higher μ means a rougher or stickier contact.

μ 的典型值从冰与冰之间约 0.01，到橡胶与干燥混凝土之间超过 1.0。μ 越大，表示接触面越粗糙或越黏。

The coefficient of friction depends on the materials and surface finish, but in the simple A-Level model it is treated as independent of speed and contact area.

摩擦系数取决于材料和表面光洁度，但在简单的 A-Level 模型中，它被视为与速度和接触面积无关。

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4. Limiting Friction and the Inequality | 极限摩擦与不等式

When an object is in static equilibrium on a rough surface, the friction force can take any value less than or equal to the limiting friction. This is expressed as:

当物体在粗糙表面上处于静力平衡时，摩擦力可以取任何小于或等于极限摩擦的值。表示为：

F ≤ μR

The equality F = μR holds only at the point of slipping, when the object is just about to move. Below this limit, friction automatically adjusts to balance the applied forces.

等式 F = μR 仅在滑动临界点即物体即将运动时成立。低于这个极限时，摩擦力会自动调整以平衡外力。

For example, a block of weight 50 N resting on a horizontal surface with μ = 0.4 has a limiting friction of 0.4 × 50 N = 20 N. If a horizontal force of 12 N is applied, the friction is also 12 N, not 20 N.

例如，一个重 50 N 的木块静止在 μ = 0.4 的水平面上，极限摩擦为 0.4 × 50 N = 20 N。如果施加 12 N 的水平力，摩擦力也是 12 N，而不是 20 N。

When drawing free-body diagrams, it is important to label friction with an arrow parallel to the surface. The direction must oppose the actual or impending relative motion.

在画受力图时，必须用平行于表面的箭头标出摩擦力。其方向必须与实际或即将发生的相对运动相反。

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5. Friction on an Inclined Plane | 斜面上的摩擦

On an inclined plane at angle θ to the horizontal, the weight mg is resolved into two components: mg sin θ parallel to the plane and mg cos θ perpendicular to the plane. The normal reaction is R = mg cos θ.

在与水平面成 θ 角的斜面上，重力 mg 分解为两个分量：平行于斜面的 mg sin θ 和垂直于斜面的 mg cos θ。法向反作用力为 R = mg cos θ。

If the block slides down a rough slope, the net force along the plane is:

如果木块沿粗糙斜面下滑，沿斜面的合外力为：

F_net = mg sin θ − μmg cos θ

The acceleration is then found from Newton’s second law:

然后根据牛顿第二定律求出加速度：

a = g(sin θ − μ cos θ)

If the block is moving up the slope, friction acts down the slope, so the net force becomes mg sin θ + μmg cos θ, and the acceleration is g(sin θ + μ cos θ) directed down the slope.

如果木块沿斜面向上运动，摩擦力沿斜面向下，因此合外力变为 mg sin θ + μmg cos θ，加速度为 g(sin θ + μ cos θ)，方向沿斜面向下。

When the block is just about to slide down, limiting friction balances the downhill weight component, giving:

当木块刚好要下滑时，极限摩擦与沿斜面向下的重力分量平衡，可得：

mg sin θ = μmg cos θ ⇒ μ = tan θ

This angle θ is known as the angle of friction. Measuring the angle at which sliding just begins provides a simple way to estimate μ.

这个角度 θ 称为摩擦角。测量物体刚开始滑动时的角度，可以

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